Scaling Approach for Estimating Stand Sapwood Area from Leaf Area Index in Five Boreal Species

Article Scaling Approach for Estimating Stand Sapwood Area from Leaf Area Index in Five Boreal species

M. Rebeca Quiñonez-Piñón 1 and Caterina Valeo 2,* 1 Geomatics Engineering, Schulich School of Engineering, University of Calgary, 2500 University Drive NW, Calgary, AB T2N 1N4, Canada; [email protected] 2 Mechanical Engineering, University of Victoria, P.O. Box 1700 STN CSC, Victoria, BC V8W 2Y2, Canada * Correspondence: [email protected]; Tel.: +1-250-721-8623

Received: 29 July 2019; Accepted: 18 September 2019; Published: 20 September 2019

Abstract: This paper presents a scaling approach for estimating sapwood area at the stand level using knowledge obtained for individual trees of ﬁve boreal species: Populus tremuloides (Michx.), Pinus contorta (Doug. ex Loud. var. latifolia Engelm.), Pinus banksiana (Lamb.), Picea mariana (Mill.) BSP, and Picea glauca (Moench) Voss. Previously developed allometric models for sapwood depth and diameter at breast height for individual tree species were used to build stand level sapwood area estimates as well as stand level leaf area estimates, in pure and mixed vascular vegetation stands. A stand’s vegetation heterogeneity is considered in the scaling approach by proposing regression models for each species. The new combined scaling approach drew strong linear correlations at the stand scale between sapwood area and leaf area using observations taken in mixed stands of Southern Alberta, Canada. This last outcome suggests a good linear relationship between stand sapwood area and stand leaf area. The accuracy of the results was tested by observing each regression model’s adequacy and by estimating the error propagated through the whole scaling process.

Keywords: sapwood area; leaf area; boreal forest; scaling approaches; allometric correlations; error propagation; Populus tremuloides; Pinus banksiana; Pinus contorta; Picea mariana; Picea glauca

1. Introduction Sapwood area supports several physiological functions, such as photosynthesis, gas ex-change, cooling, nutrient transport, and transpiration. Also, sapwood area has been theoretically related to leaf area with Shinozaki’s pipe model theory [1]. The pipe model theory is supported by several studies, most of which modeled species specific leaf area (LAsp) versus sapwood area (SAsp) to estimate a single tree’s leaf area (where the tree’s sapwood area is the predictor). Linear models for about 20 conifers, such as Scots pine (Pinus sylvestris L.) [2]; Ponderosa pine (Pinus ponderosa Douglas ex Lawson) [3]; Loblolly pine (Pinus taeda L.) [4]; Douglas fir (Pseudotsuga menziesii (Mirbel) Franco) [3,5,6]; Lodgepole pine (Pinus contorta)[7–10]; Engelmann spruce and Subalpine fir (Picea engelmanni Parry ex Engelmann and Abies lasiocarpa (Hooker) Nuttall) [10]; Pinyon pine (Pinus edulis Engelmann) and One-seeded Juniper (Juniperus monosperma (Engelmann) Sargent) [11]; and Balsam fir (Abies balsamea (L.) Mill.) [12,13], were reported. While deciduous species haven’t been studied as greatly as coniferous species, there are some published works for Mountain ash (Eucalyptus regnans F. Muell.) [14]; Trembling aspen (Populus tremuloides)[10]; and Cherry bark oak (Quercus falcate Q.Pagoda) and Green ash (Fraxinus pensylvanica Marshall) [15]. In this last work, the model for predicting leaf area was improved by not only having sapwood area as a predictor, but by adding total height and live crown ratio to the model [15]. Researchers studying the same species under different site conditions have reported different LAsp:SAsp regression models (linear and non-linear). The discrepancy between results have helped to

Forests 2019, 10, 829; doi:10.3390/f10100829 www.mdpi.com/journal/forests Forests 2019, 10, 829 2 of 19

understand that the LAsp:SAsp relationship is driven by site conditions such as stand density [16–18], climatic factors [19], and physical characteristics [20]. Naturally, it is expected that the LAsp:SAsp relationship is species-specific [20], but this has not always been observed. For example, the Lodgepole pine (Pinus contorta) LAsp:SAsp allometric relationship is linear [21] but in another study, a nonlinear regression better explained this relationship [10]. Thus, while in theory the LAsp:SAsp relationship is positioned as linear, this could change due to site conditions. Average sapwood depth (sd) has been used instead of sapwood area to indirectly estimate other forest stand characteristics such as canopy cover densities and foliage biomass, by combining remotely sensed data and field data [22,23]. The method consists of estimating foliage biomass as a function of the diameter at the breast height (D) by previously modelling sd:D relationships to predict D for a whole tree stand [23]. The authors used equations to estimate foliage biomass and sapwood area. A previously reported relationship for sd:D to estimate foliage biomass was used as an indicator of D in [22]. The sd:D relationship used in both studies was obtained from [24]. To the authors’ knowledge, [24] is one of a very small number of published works reporting results for single tree sd:D relationships ([24] cited three more published works on this topic). In Douglas fir (Pseudotsuga menziesii) and some other conifers, sapwood depth increases as the inside bark tree’s diameter increases [24]. Most of the species showed that trees had an exponential sapwood depth growth until they reached a diameter of 25.4 38.1 cm, where sapwood depth would then plateau. Sapwood depth in older trees − tended to increase at a slower rate than younger trees. Except for Pinus contorta, this was the general tendency, where there was large variability in sapwood depth, and some trees that were smaller in diameter showed a larger sapwood depth than those with a larger diameter. Douglas fir trees with the same diameter had different sapwood depths according to their location (coast area or interior land), and elevation [24]. Even though there is sufficient work supporting the strong relationships between a single tree’s sapwood area and leaf area, these models cannot be used to scale up to large regions or stands. In a forested region, it is expected that leaf area increases as the ground area increases, and of course as the ground area increases, the greater likelihood of the area having multiple species. Therefore, it is insufficient to use a single tree model to interpolate either leaf area or sapwood area values for a group of trees composed of different species [25,26]. Thus, it is necessary to develop species specific models for the LAsp:SAsp relationship. More recent studies attempted to model Eucaliptus regnans’ stand sapwood area-basal area ratios by scaling up individual stumps’ visual heartwood-sapwood differentiation and using digital photography. They obtained a linear model with a coefficient of determination of 0.85 [27]). Modeled sapwood area at the stand level using LIDAR images and individual tree detection algorithms were used to predict sapwood area/basal area relationships at the stand level [28]. Their predicted values drew correlation coefficients that varied from 0.5 to 0.84, where the most adequate fit was with a regression model that combined LIDAR derived data and observed basal area. A mathematical model of a catchment’s basal area-sapwood area was created by indirect estimates through LIDAR imaging [29]. Not all methods for estimating sapwood area will give the same results [30,31] and the direct or indirect measurement of leaf area may also produce different outcomes [32]. Given this, the research hypothesis in this study is that in addition to site conditions—such as stand density, climatic factors, and physical characteristics, the methods used to measure and estimate sapwood area and leaf area, significantly influence the LAsp:SAsp relationship and therefore, the calculation of the error propagated to the final estimate will determine the quality of an allometric model and help detect if the model contributes to over- or under-estimates. A thorough statistical analysis to determine the normality of each dataset, will help to determine if a regression model fits the dataset or whether other statistical/mathematical models should be considered. Hence, the objectives of this paper are: (1) To create reliable regression models—if adequate for each dataset—to estimate sapwood area at the breast height for Populus tremuloides, Pinus banksiana, Pinus contorta, Picea glauca, and Picea mariana; (2) to develop appropriate scaling LA-SA relationships for forest stands comprised of a mixture of these Forests 2019, 10, 829 3 of 19 species; and (3) to determine the absolute error propagated while scaling sapwood depth from an individual tree up to the stand level.

2. Materials and Methods

2.1. Model Approach and Sampling Design For scaling purposes (from tree-to-stand level), a vegetated stand with vascular species has been conceptualized as an area of forested land (a stand) with a single tree with a sapwood area SAplot equal to the summation of all the individual trees’ sapwood cross-sectional area (SAi) inside the stand.

Xn SAplot = SAi (1) i = 1 where SAi is species speciﬁc for each tree i in the stand containing n trees of diﬀerent species. Similarly, the single tree’s leaf area (LAplot) will be the summation of all trees’ leaf area inside the stand.

Xn LAplot = LAi (2) i = 1

Despite the simplicity of the concept, it considers vegetation heterogeneity by using species speciﬁc models to scale up biophysical characteristics. At the tree-level, linear regression models were developed for each species sapwood depth sd versus outer bark diameter at breast height (DOB) data so that Equation (3) below could be used to estimate SAi:

2 SA = π D sd sd (3) i OBi i − i

th th where sdi is each i tree’s average sapwood depth and DOBi is the i tree’s diameter at breast height. Equation (3) calculates SAi as the region lying between two concentric circles within a tree’s cross-section. The outermost circle borders the bark and vascular cambium, while the innermost one bounds the heartwood. These circles are naturally irregular but tree trunks are treated as having a cylindrical shape. The models obtained were used to estimate every tree’s sapwood area SAi inside the delimited stand. Equation (1) is then used to estimate the stand level sapwood area (SAplot) of that single tree representing the whole stand. The correlation between species specific leaf area and sapwood area (LAsp:SAsp) and a mathematical scaling approach detailed in Section3 were used to estimate LAplot:SAplot. The leaf area for the stand LAplot, was obtained by measuring the Leaf Area Index (LAI) by light transmission, and the surface ground area of interest. Stands of four different sizes are used to validate the LAsp:SAsp relationship. Four stands were 60 60 m, and nine were 10 10 m. These stands were × × located in the Sibbald Areas of Kananaskis Country, Alberta, Canada. The species composition of these stands was either dominated by deciduous trees (mainly Populus tremulolides), or coniferous trees (Pinus contorta and Picea glauca). Figures1 and2 show the location and delimitation of these stands. Two more stands were delimited at Whitecourt, in northern Alberta: One was dominated by Pinus banksiana individuals (20 20 m) and one was dominated by Picea mariana (15 15 m) × × individuals. The stands whose species composition was dominated by a conifer tree were labelled as “coniferous”, and for those whose species composition was dominated by a deciduous tree were labeled as “deciduous”. The 10 10 m stands were distinguished from the larger ones ( 300 m2) by × ≥ adding a prime symbol in front of their assigned number. Field data collected at each stand included: (a) Number of trees per stand, (b) species identification, and outer bark diameter at breast height for every tree inside the stand, and (c) Leaf Area Index for the stand. LAI was measured in the 60 60 m × stands using the Tracing Radiation and Architecture Canopies (TRAC, 3rd Wave Engineering Co.; Forests 2019, 10, 829 4 of 19

Nebraska, US) was used to measure LAI in the stands of 20 × 20 m and 15 × 15 m, located in Whitecourt. Kananaskis Valley is a Montane closed forest formation [34] within the Rocky Mountains [35,36]. The Montane forest is classified as an ecoregion within the Cordilleran eco-province with a particular mix of physiography and air masses leading to unique climatic conditions [36]. Within Alberta, the Montane forest maintains the warmest temperatures during the winter than any other forested ecosystem. This type of forest has ridged foothills and a marked rolling topography. The Whitecourt forest is within the mid boreal mixed-wood ecoregion [36] and considered a closed forest formation [37] in the Southern Alberta uplands [36]. Further details on the field sites can be found in [32,38].

2.2. Treatment of Saplings Since saplings generally lack heartwood and being mostly composed of sapwood [21,31,39], saplings correlations between DOB and 𝑠𝑑, or between DOB and cross-sectional area per species (SAsp) will be different than for mature trees and thus, were treated separately. Furthermore, due to their size and composition, saplings were considered part of the understory, and this study focused on Forestsscaling2019 allometric, 10, 829 correlations of the overstory. Saplings were considered those trees with4 ofDOB 19 ranging between 2.41 cm and 10.2 cm, and heights between 38.1 cm and 76.2 cm. Trees found inside of the stands with a DOB ≤ 10cm were considered saplings, and were excluded from all allometric Nepean, Ontario, Canada) device. The Canopy Analyzer LAI-2000 (LI-COR Incorporated; Lincoln, correlations. Nebraska, US) was used to measure LAI in the stands of 20 20 m and 15 15 m, located in Whitecourt. × ×

Forests 2019, 10, 829 5 of 19 Figure 1.1. GeographicalGeographical location location of of coniferous coniferous plots. plots. The plotsThe plots are in are the in Sibbald the Sibbald area, Kananaskis area, Kananaskis Country. Country.

. Figure 2. Geographical location of deciduous plots. The plots are in the Sibbald areas, south-east of BarrierFigure Lake 2. Geographical (Kananaskis location Country). of deciduous plots. The plots are in the Sibbald areas, south-east of Barrier Lake (Kananaskis Country).

2.3. Statistical Analysis For each species, we performed statistical analyses to determine the most adequate regression model. These statistics include the Pearson’s correlation coefficient, detection of outliers, the regression analysis, ANOVA, list of unusual observations, and the lack-of-fit test. Finally, each linear model’s residuals were examined to check the model adequacy. Model adequacy checking included a normal plot of residuals and a plot of residuals versus fitted values. In addition to the correlation analysis, a pairwise comparison of the coefficient of variation (COV) confidence intervals (C.I.) was used as an indication of the relationship between the scaling parameters at the stand scale (i.e., between SAplot and LAI, and between SAplot and LAplot). The pairwise comparison consisted of comparing the parameters’ C.I.s to see if they overlapped or not. This was another way to test if there was correlation between the variables. Because they are different parameters and the units differ, it was not suitable to use mean values or standard deviations to test the similarity between the two sample populations [40–42]. For 100 (1 − α)% and ν = n − 1 degrees of freedom, the modiﬁed McKay confidence interval for a COV [41] is:

. . −1 𝐾 + 𝐾≤𝐶𝑂𝑉≤𝐾 −1 𝐾 + (4)

𝑆 where K is the point estimate of COV (𝐾= ), u1 = 𝜒 and u2 = 𝜒 . Therefore, K is a 𝑋 ,/ ,/ function of the sample standard deviation (S), and the sample mean (𝑋). Another modified approach to MacKay C.I. is the suggested by [43]:

1+𝐾 . 1+𝐾 . 𝜒 ≤𝐶𝑂𝑉≤𝜒 (5) ,/ 𝑛𝐾 ,/ 𝑛𝐾

Forests 2019, 10, 829 5 of 19

Kananaskis Valley is a Montane closed forest formation [33] within the Rocky Mountains [34,35]. The Montane forest is classiﬁed as an ecoregion within the Cordilleran eco-province with a particular mix of physiography and air masses leading to unique climatic conditions [35]. Within Alberta, the Montane forest maintains the warmest temperatures during the winter than any other forested ecosystem. This type of forest has ridged foothills and a marked rolling topography. The Whitecourt forest is within the mid boreal mixed-wood ecoregion [35] and considered a closed forest formation [36] in the Southern Alberta uplands [35]. Further details on the ﬁeld sites can be found in [31,37].

2.2. Treatment of Saplings Since saplings generally lack heartwood and being mostly composed of sapwood [20,30,38], saplings correlations between DOB and sd, or between DOB and cross-sectional area per species (SAsp) will be different than for mature trees and thus, were treated separately. Furthermore, due to their size and composition, saplings were considered part of the understory, and this study focused on scaling allometric correlations of the overstory. Saplings were considered those trees with DOB ranging between 2.41 cm and 10.2 cm, and heights between 38.1 cm and 76.2 cm. Trees found inside of the stands with a D 10cm were considered saplings, and were excluded from all allometric correlations. OB ≤ 2.3. Statistical Analysis For each species, we performed statistical analyses to determine the most adequate regression model. These statistics include the Pearson’s correlation coefficient, detection of outliers, the regression analysis, ANOVA, list of unusual observations, and the lack-of-fit test. Finally, each linear model’s residuals were examined to check the model adequacy. Model adequacy checking included a normal plot of residuals and a plot of residuals versus fitted values. In addition to the correlation analysis, a pairwise comparison of the coefficient of variation (COV) confidence intervals (C.I.) was used as an indication of the relationship between the scaling parameters at the stand scale (i.e., between SAplot and LAI, and between SAplot and LAplot). The pairwise comparison consisted of comparing the parameters’ C.I.s to see if they overlapped or not. This was another way to test if there was correlation between the variables. Because they are different parameters and the units differ, it was not suitable to use mean values or standard deviations to test the similarity between the two sample populations [39–41]. For 100 (1 α)% and ν = n 1 degrees of freedom, the modified McKay confidence interval for a − − COV [40] is: u + 2 u 0.5 u + 2 u 0.5 1 1 K2 + 1 − K COV K 2 1 K2 + 2 − (4) n − ν ≤ ≤ n − ν where K is the point estimate of COV (K = S ), u = χ2 and u = χ2 . Therefore, K is a function X 1 ν,1 α/2 2 ν,α/2 − of the sample standard deviation (S), and the sample mean (X). Another modified approach to MacKay C.I. is the suggested by [42]:

" # 0.5 " # 0.5 1 + K2 − 1 + K2 − χ2 COV χ2 (5) ν,1 α/2 2 ν,α/2 2 − nK ≤ ≤ nK

3. Model Implementation and Data Results For all five species, the tree-level sapwood depth was measured by microscopical analysis of wood anatomy to differentiate between sapwood and heartwood, as well as to differentiate sapwood from cambium and phloem. The sampling, sapwood depth, average sapwood depth, and DOB results obtained for each species sampled to create the tree-level allometric correlations reported in [31]. Forests 2019, 10, 829 6 of 19

3.1. Tree-Level Allometric Correlations

Pinus banksiana and Pinus contorta average sapwood depth sd was not well correlated with their respective DOB [31]. This was also true of Picea mariana. Picea glauca and Populous tremuloides did however, show signiﬁcant, positive linear correlations between sd and DOB. For Picea glauca, the relationship from [31] and used here was:

sd = 0.09D 0.70 R2 = 0.84 (6) OB − For Populous tremuloides the relationship obtained in [31] was used and is as follows:

2 sd = 0.24DOB + 0.92 R = 0.52 (7)

Depending on the correlation between sd and DOB (that is, whether there is no relationship or a signiﬁcant linear relationship like those shown in Equations (6) and (7)), the methods for scaling up to the stand level using DOB required a species-based approach. Thus, for Picea glauca and Populous tremuloides, Equations (6) and (7) were used to determine stand level sd estimates from stand level estimates of DOB, respectively. For those species without a strong tree-level linear correlation, a simple stand average, i.e., sd, was proposed.

3.2. Stand-Level Allometric Correlations

3.2.1. Scaling Up Sapwood Area from the Tree to the Stand Level Since it was feasible to fit a linear regression model for two of the five studied species, the aggregation of cross-sectional sapwood area to the stand level to obtain SAplot was performed by fusing two approaches. To estimate the sapwood area at every stand, if Picea glauca and/or Populus tremuloides trees were present, their respective linear model was applied—this was the first scaling approach. But under the presence of Pinus banksiana and/or Pinus contorta and/or Picea mariana, their respective average sapwood area was applied, which is the second scaling approach. Therefore,

XL XM SAplot = SAj + SAk (8a) j=1 k=1 which is further expanded as (8b):

L Nsp,j M X X 2 X SA = D sd0 sd0 π + SA N (8b) plot OBi,j i,j − i,j k sp,k j=1 i=1 k=1 where L is the number of species present with a linear model (i.e., L = 0, 1 or 2 depending on whether none, one, or both are present of Picea glauca or Populus tremuloides; thus, if L = 0, the summation with j th beginning at 1 is null), Nsp,j is the total number of trees inside the stand of species j; DOBi,j is the i 0 th individual’s DOB of species j; and sdi,j is the i individual’s estimated sapwood depth for species j, and estimated for trees of species Picea glauca with Equation (6), or Populus tremuloides with Equation (7). The second term on the right-hand side of Equation (8a) pertains to those species that did not produce a signiﬁcantly linear model. Thus, M is the number of species k present in the stand with k being Pinus banksiana, Pinus contorta, or Picea mariana. M = 0, 1, 2 or 3; and Nsp,k is the number of trees in the stand of each of these species. Based on previous results [31], it was assumed that the sapwood cross-sectional area for each of these three species remained relatively constant as the tree grew. Therefore, for each of these species k, the average sapwood area was calculated based on results from [31] with important details given in the next paragraph for SAk. Forests 2019, 10, 829 7 of 19

The study of [31] determined what relationships existed for these five boreal species between sd and DOB. Microscopically differentiated average sapwood depth (averaged over the four cardinal directions) from wood cores obtained for the fives species was correlated with DOB for each tree in that study. Average sapwood area for each species that did not have a linear relationship between sd and DOB was computed using Equation (9): Pt 2 k D sd sd π i=1 OBi,k i,k − i,k SAk = (9) tk where, sdi,k is the average sapwood depth for each cored tree i, of species k, in that study, and accordingly,

DOBi,j was the outer bark diameter at breast height for tree i of species k and tk was the number of cored trees in the sample sets of [31]. Values of SAk and tk for all ﬁve species are reported in Table1 and [ 31] provides greater details not contained here.

Table 1. Values of average tree-level sapwood area for each species computed with data from [31].

Only for the ﬁrst three species on the list, SAk was used to estimate stand level (SA) in Equation (8b).

2 Species (k) tk SAk (cm ) Pinus banksiana 33 176.47 and Pinus contorta Picea mariana 21 213.57 Populus tremuloides 26 277.13 Picea glauca 17 277.52

The SA calculated for each 60 60 m and 10 10 m stand, including tree species, tree quantity, plot × × stand DOB statistics and the error associated with SAplot, are given in Tables2 and3, respectively. The conifer sites were those sites containing Picea glauca, Pinus contorta, Pinus banksiana, and Picea mariana. The deciduous sites were mainly composed of Populus tremuloides. If deciduous individuals were present in the coniferous stands, their numbers counted for less than 10% of the total tree quantity or were mainly saplings. Such was the case of the site Conifer-5, with a Populus tremuloides tree quantity of 114; however, 92 individuals were saplings. Thus, the count of deciduous trees inside the stand was a small portion of the total in terms of sapwood area contribution. Most of the deciduous sites were 100% pure stands, and if any other tree was present, it counted for less than 5% of the total stand’s tree quantity (e.g., Deciduous-6). Most of the coniferous sites (60 60 m and 10 10 m) were composed of × × Picea glauca and Pinus contorta trees. The two pure coniferous sites were Conifer-11 and Conifer-12.

Table 2. Descriptive statistics of the 60 60 m stands located in the Sibbald areas of Kananaskis Country, × Alberta, and Whitecourt, Alberta. Tree quantity is either Nsp,k or Nsp,j depending on the species; SAsp is either SAj or SAk in Equation (8b), depending on the species, ∆SAplot is the absolute error associated 2 with SAplot. DOB units are in cm, and all areas are in units of m .

No. of Max. Min. Avg. Plot Tree species SAsp SAplot ∆SAplot Trees DOB DOB DOB Picea glauca 1 434 43.3 2.2 13.14 2.92 Conifer-4 Pinus contorta 276 33.42 5.73 20.15 4.87 7.79 0.03 ± Picea glauca 164 48.38 2.86 18.31 2.29 Conifer-5 1 Pinus contorta 48 26.1 1.59 14.09 0.85 Populus tremuloides 114 19.42 2.86 7.23 0.14 3.28 0.03 ± Conifer-11 2 Pinus banksiana 129 25.15 6.37 16.57 2.28 2.28 0.03 ± Conifer-12 2 Picea mariana 60 23.55 5.09 15.69 1.28 1.28 0.01 ± Deciduous-1 1 Populus tremuloides 83 25.46 9.55 18.57 1.45 1.45 0.01 ± Populus tremuloides 1 498 31.19 8.91 19.1 9.37 Deciduous-6 Picea glauca 14 48.38 7 23.4 0.18 9.55 0.08 ± 1 Indicates Sibbald region of Kananaskis Country and 2 indicates Whitecourt region. Forests 2019, 10, 829 8 of 19

Table 3. Descriptive statistics of the 10 10 m stands (indicated with a prime symbol) located in the × Sibbald areas of Kananaskis Country, Alberta. Note, ∆SAplot is the absolute error associated with SAplot. The error propagation estimation is detailed in Section 3.3. DOB units are in cm, and all areas are in units of m2.

No. of Max. Min. Avg. Plot Tree Species SAsp SAplot ∆SAplot Trees DOB DOB DOB Picea glauca 29 24.51 5.091 6.53 0.16 Conifer-1’ Pinus contorta 12 26.74 14.01 19.89 0.21 0.37 0.001 ± Pice glauca 15 27.06 7.641 14.88 0.21 Conifer-2’ Pinus contorta 9 27.37 16.23 20.23 0.16 0.37 0.001 ± Picea glauca 19 25.78 4.461 12.48 0.19 Conifer-3’ Pinus contorta 13 29.6 10.82 19.98 0.23 0.42 0.0007 ± Picea glauca 9 21.33 3.501 12.59 0.04 Conifer-4’ Pinus contorta 14 27.06 11.78 20.35 0.25 0.29 0.0003 ± Picea glauca 4 28.33 12.41 18.94 0.05 Conifer-5’ Pinus contorta 13 32.79 15.92 22.33 0.23 0.28 0.0002 ± Picea glauca 5 33.74 5.091 16.87 0.19 Conifer-10’ Pinus contorta 14 34.7 12.73 21.99 0.11 Populus tremuloides 1 21.01 0.02 0.32 0.0005 ± Deciduous-7’ Populus tremuloides 31 32.79 9.871 17.15 0.34 0.34 0.004 ± Deciduous-8’ Populus tremuloides 28 28.01 14.01 20.29 0.59 0.59 0.005 ± Deciduous-9’ Populus tremuloides 22 23.24 14.64 19.92 0.43 0.43 0.004 ±

3.2.2. Leaf Area Estimates at the Stand-Level Leaf Area Index, LAI, was measured in the 60 60 m stands with the Tracing Radiation and × Architecture Canopies (TRAC, 3rd Wave Engineering Co.; Nepean, Ontario, Canada) device. At the two stands located in Whitecourt (Conifer-11, Conifer-12), the Effective Leaf Area Index, LAIeff, was measured with the Canopy Analyzer LAI-2000 (LI-COR Incorporated; Lincoln Nebraska, US. To convert LAIeff to Leaf Area Index (LAI), we applied Chen’s equation [43]:

LAI = (1 α )LAI γEΩE (10) − l eff where αl is the woody-to-total area ratio (i.e., tree’s woody fraction), γE is the needle-to-shoot area ratio (i.e., fraction of needles per shoot), ΩE is the clumping index. In order to derive the αl and γE values from the typical values reported by [44], the age and productivity of the stand must be known. Thus, age and productivity characteristics were defined by comparing the LAIeff with typical values reported in [45]. The LAIeff of the Pinus banksiana stand was similar to an intermediate/medium productivity stand (LAIeff = 2.20) with αl of 0.05 and γE of 1.35. The Picea mariana stand LAIeff value was similar to a mature medium/high productivity stand (LAIeff = 2.78), thus, αl is assumed as 0.14 and γE as 1.35. Finally, ΩE was derived from typical values reported by [45] as well; thus, the ΩE values for Picea mariana and Pinus banksiana were respectively set as 0.65 and 0.75. The fifteen stands estimated LAI (LAIplot) concur with the previous reported values by [43,45,46]. The total leaf area per stand, LAplot was estimated based on the definition of Leaf Area Index: “The total one-sided (or one half of the total all-sided) green leaf area per unit ground surface area”:

LAplot = (LAI)Aplot (11)

Table4 displays the estimates of LAI . The coniferous 10 10 m stands’ areas were adjusted and plot × reported as 150 m2 in order to take into account what is known as the “TRAC footprint”. The TRAC footprint was created by trees with shadows large enough to fall into the delimited stand influencing the TRAC measurements. Thus, the LAI measured in the field belonged to a larger area than the delimited one. This effect was not evident in the 10 10 m deciduous stands. Furthermore, the × TRAC footprint was influenced by tree height and solar zenith angle (θ)[44]. Therefore, the TRAC footprint was adjusted by assuming that in Kananaskis Country, the coniferous trees’ average height was approximately 15 m, and the solar noon zenith angle was about 45.21◦ on the date when LAI was Forests 2019, 10, 829 9 of 19 measured (day 238 of the year). Therefore, the extent of the footprint (tree height tanθ) was 15 m. × This extent would occur in only one side of the stand giving an LAI for a stand area of 10 15 m. ×

Table 4. Leaf Area Index (LAI) and estimated LAplot according to stand size where ∆LAplot is the absolute error associated with LAplot. The error propagation estimation is detailed in Section 3.3. Prime symbol indicates 10 10 m stands. No prime symbol indicates 60 60 m stands. × × 2 2 2 Site LAI Plot Size (m ) LAplot (m ) ∆LAplot (m ) Conifer-1’ 6.90 150 1035.00 19.90 ± Conifer-2’ 6.04 150 906 19.04 ± Conifer-3’ 6.57 150 985 19.57 ± Conifer-4’ 5.34 150 801 18.34 ± Conifer-5’ 4.51 150 676.5 17.51 ± Conifer-10’ 6.12 150 918 19.12 ± Conifer-4 5.71 3600.00 20,556.00 1338.20 ± Conifer-5 2.54 3600.00 9144.00 855.10 ± Conifer-11 3.76 400 1502.52 259.32 ± Conifer-12 4.96 300 1486.97 242.82 ± Deciduous-1 2.64 415 1093.53 405.85 ± Deciduous-6 3.14 3600,00 11,304.00 1144.27 ± Deciduous-7’ 2.30 100 230 12.30 ± Deciduous-8’ 3.57 100 357 13.57 ± Deciduous-9’ 3.22 100 322 13.22 ±

3.2.3. Stand Level Leaf Area Sapwood Area Allometric Correlations In the case of the Populus tremuloides sites, a close similarity was found between the two sets of stands of 10 10 m and 60 60 m in terms of LAI. In order to support this observation, the equality of × × variances with Bartlett’s test and the F-test was conducted. Based on the two statistical test results, the LAI variances of the two sample sets were equal (C.I. = 95%) and the two sets were merged to observe the correlation of Populus tremuloides leaf area and sapwood area at the stand level. The explanation for such a similarity between Populus tremuloides’ leaf area at different scales may be related to the canopy type. Unlike coniferous trees, Populus tremuloides canopy is horizontally wide, creating small gaps between the canopy of neighboring trees. This effect of Populus tremuloides canopies is due to their wide-circular leaves and their alternate (not clumped) distribution. At the same time, the leaves are predominantly at the top of the tree, creating a rounded crown with a large diameter. Moreover, in the studied sites, the Populus tremuloides tree heights were all similar. The combination of these characteristics gives wider tree shadows that practically cover the entire stand’s floor at noon. Thus, inside the stand there is little room for observing the footprint of external trees in any direction meaning that the TRAC device mostly measures the LAI inside the stand with a negligible footprint. This was proven by examining the LAI measured in the east–west and north–south transects. In both east–west and north–south directions, LAI was observed to have practically the same value (a paired T-test proved that the difference of the population mean was equal to zero with an α = 0.05). Hence, it was concluded that it was possible to merge both data sets for use in a linear regression model. For all Populus tremuloides sites, there was a strong linear correlation between SAplot and LAplot (ρ 0.999), and the p-value that equaled zero gave sufficient evidence to conclude that the correlation ≈ was not zero (α = 0.05). Results of the regression analysis supported the decision of fitting a linear model to the data, showing that LAplot resulted in a significant predictor of SAplot with a p-value < 0.0001 (α = 0.05). The ANOVA results determined that LAplot contributes significantly to the model (α = 0.05). The first fitted model (Figure3) had a high coe fficient of determination (R2 = 99.8%) that 2 supported the theory of SAplot being fairly well explained by a linear model. The R adj (99.8%) was 2 2 2 close in value to the R . The R pred (57.77%) was weak with a large difference from R adj. Thus, there was a slight indication of one value inflating the prediction, and thus, this model might not be suitable for other stands. The list of unusual observations draws attention to the large influence Forests 2019, 10, 829 10 of 19 combination of these characteristics gives wider tree shadows that practically cover the entire stand’s floor at noon. Thus, inside the stand there is little room for observing the footprint of external trees in any direction meaning that the TRAC device mostly measures the LAI inside the stand with a negligible footprint. This was proven by examining the LAI measured in the east–west and north– south transects. In both east–west and north–south directions, LAI was observed to have practically the same value (a paired T-test proved that the difference of the population mean was equal to zero with an α = 0.05). Hence, it was concluded that it was possible to merge both data sets for use in a linear regression model. For all Populus tremuloides sites, there was a strong linear correlation between SAplot and LAplot (ρ ≈ 0.999), and the P-value that equaled zero gave sufficient evidence to conclude that the correlation was not zero (α = 0.05). Results of the regression analysis supported the decision of fitting a linear model to the data, showing that LAplot resulted in a significant predictor of SAplot with a p-value < 0.0001 (α = 0.05). The ANOVA results determined that LAplot contributes significantly to the model (α = 0.05). The first fitted model (Figure 3) had a high coefficient of determination (R2 = 99.8%) that supported the theory of SAplot being fairly well explained by a linear model. The R2adj (99.8%) was close in value to the R2. The R2pred (57.77%) was weak with a large difference from R2adj. Thus, there Forests 2019, 10, 829 10 of 19 was a slight indication of one value inflating the prediction, and thus, this model might not be suitable for other stands. The list of unusual observations draws attention to the large influence that the Deciduous-6that the Deciduous-6 site gives site to gives the tomodel, the model, considerin consideringg it was it was an anunusual unusual observation. observation. Even Even if if the observation was was not not considered considered an an outlier, outlier, it was removed from the sample set, and the regression model was fitted fitted with only the four observations. In In the the second second regression regression attempt (Figure 44),), allall thethe coefficientscoefficients of determination values were lo lowerwer than the ones of thethe firstfirst model.model. The R2 was high 22 22 (99.2%), and thethe RR pred (79.5%)(79.5%) was was in in reasonable reasonable agreement with thethe RR adj (98.78%).(98.78%). Thus, Thus, the the second second model was proposed for practical use.

SA = 0.2943 + 0.000805 LA 8 ) 2

S 0.191622 R-Sq 99.8%

Sapwood area Area (m Area area Sapwood 2 R-Sq(adj) 99.8%

Regression 95% CI 0 0 2000 4000 6000 8000 10000 12000 Leaf Area (m2)

Figure 3. Fitted linear regression model for Populus tremuloides predicted SAplot in relation to its LAplot. ForestsFigure 2019, 10 3., 829Fitted linear regression model for Populus tremuloides predicted SAplot in relation to its LAplot11. of 19 This model includes all 60 60 m and 10 10 m stands. This model includes all 60 × 60 m and 10 ×× 10 m stands. 1.8

1.6 SA = 0.06649 + 0.001270 LA ) 2 1.4

1.2

1.0

0.8

0.6 S 0.0562600 R-Sq 99.2%

Sapwood area Area (m Area areaSapwood 0.4 R-Sq(adj) 98.8% Regression 0.2 95% CI 0.0 200 300 400 500 600 700 800 900 1000 1100 Leaf Area (m2)

Figure 4. Fitted linear regression model for Populus tremuloides predicted SAplot in relation to its LAplot. Figure 4. Fitted linear regression model for Populus tremuloides predicted SAplot in relation to its LAplot. This model does not include the stand Deciduous-6. This model does not include the stand Deciduous-6.

The coniferous SAplot and LAplot estimates for the 10 10 m and 60 60 m stands are given in The coniferous SAplot and LAplot estimates for the 10 × 10× m and 60 × 60 m× stands are given in Table Table5. For all coniferous sites, there was a strong linear correlation between the SAplot and the LAplot, 5.and For the allp -valueconiferous supports sites, that there the was correlation a strong islinear not zero. correlation However, between if a regression the SAplotmodel and the was LA derivedplot, and thebyusing P-value both supports sample that sets (10the correlation10 m and 60is not60 zero m stands),. However, the lack-of-fit if a regression test is model significant was atderived a p-value by × × usingof 0.019. both The sample lack-of-fit sets (10 test × suggested10 m and 60 a possible× 60 m stan curvatureds), the inlack-of-fit the model test and is significant that some at other a p-value type of 0.019. model The should lack-of-fit be fitted. test suggested It was assumed a possible that curvature the mismatch in the between model and the that two some data setsother was type due of model should be fitted. It was assumed that the mismatch between the two data sets was due to the overestimation of LAI due to the influence of the footprint at the 10 × 10 m scale. Therefore, the sapwood area for the 10 × 10 m stands were underestimated between 20% and 29.6%. Hence, in the case of the coniferous sites, the obtained values for the 10 × 10 m stands were not suitable for combination with the 60 × 60 m stands because of the footprint caused by the canopy type (which was not randomly distributed, that is, it was clumped, and had large open areas that allow trees outside of the stand to reflect their shadows inside of it).

Table 5. Estimated sapwood area and their respective leaf area per stand, conifer sites. The first six sites are of size 10 × 10 m while the last four are of 60 × 60 m.

Pearson’s Correlation Site SAplot (m2) LAplot (m2) Coefficient, (P-value) Conifer-1’ 0.37 1,035 Conifer-2’ 0.37 906 Conifer-3’ 0.42 985.5 Conifer-4’ 0.29 801 Conifer-5’ 0.28 676.5 Conifer-10’ 0.33 918 0.851, (0.033) Conifer-4 7.79 20,556.00 Conifer-5 3.28 9,144.00 Conifer-11 2.28 1,502.52 Conifer-12 1.28 1,486.97 0.982, (0.022)

Forests 2019, 10, 829 11 of 19 to the overestimation of LAI due to the inﬂuence of the footprint at the 10 10 m scale. Therefore, × the sapwood area for the 10 10 m stands were underestimated between 20% and 29.6%. Hence, in × the case of the coniferous sites, the obtained values for the 10 10 m stands were not suitable for × combination with the 60 60 m stands because of the footprint caused by the canopy type (which was × not randomly distributed, that is, it was clumped, and had large open areas that allow trees outside of the stand to reﬂect their shadows inside of it).

Table 5. Estimated sapwood area and their respective leaf area per stand, conifer sites. The ﬁrst six sites are of size 10 10 m while the last four are of 60 60 m. × × 2 2 Site SAplot (m ) LAplot (m ) Pearson’s Correlation Coeﬃcient, (p-Value) Conifer-1’ 0.37 1035 Conifer-2’ 0.37 906 Conifer-3’ 0.42 985.5 Conifer-4’ 0.29 801 Conifer-5’ 0.28 676.5 Conifer-10’ 0.33 918 0.85 1, (0.033) Conifer-4 7.79 20,556.00 Conifer-5 3.28 9144.00 Conifer-11 2.28 1502.52 Forests 2019Conifer-12, 10, 829 1.28 1486.97 12 of 19 0.98 2, (0.022) 3 0.970.97,(<3, 0.001)(<0.001) 1 2 3 1 Correlation between 10 10 m stands,2 correlation between 60 60 m stands, correlation3 includes all stands. Correlation between 10 × 10× m stands, correlation between 60× × 60 m stands, correlation includes all stands

The sample set was then divided intointo thethe 1010 × 10 m and the 60 × 60 m stands to fit fit two separate × × regression models models for for the the conifer conifer stands. stands. The The first first model model corresponded corresponded to the to 10 the × 10 10 m stands10 m (Figure stands × (Figure5), and 5the), and latter the model latter was model fit for was the fit 60 for × the60 m 60 stands60 m (Figure stands 6). (Figure Regression6). Regression analysis analysisand ANOVA and × ANOVAresults suggest results that suggest the LA thatplot is the a LAsignificantplot is a significantpredictor of predictor the SAplot of (α the = 0.05).SAplot However,(α = 0.05). the However, model’s 2 2 theR2pred model’s for theR 10pred × for10 m the stands 10 10 denote m stands inadequacy denote inadequacy for future forpredictions, future predictions, and the R and2pred the (38.37%)R pred × 2 (38.37%)significantly significantly differs from diff ersthe fromR2adj. theTheR regressionadj. The regression model for model the 60 for × 60 the m 60 stands,60 m however, stands, however, show a × showbetter agreement a better agreement between its between coefficients its coe offfi determcientsination of determination (Figure 6). Still (Figure the 6difference). Still the between di fference R2adj 2 2 betweenand the RR2predadj wasand thelargeR pred(68.60%),was large but the (68.60%), model but adequacy the model check adequacy gave enough check gave evidence enough to evidencesupport tothe support decision the for decision fitting a for linear fitting model a linear to the model 60 × 60 to them data 60 set.60 mBoth data models set. Both have models similar have slopes similar that 2 × slopesdiffer by that just di ffabouter by 0.37 just cm about2. 0.37 cm .

0.45 SA = 0.03406 + 0.000349 LA ) 2 0.40

0.35

0.30

S 0.0317959 R-Sq 71.8% Sapwood area Area (m Area area Sapwood 0.25 R-Sq(adj) 64.7%

Regression 95% CI 0.20

700 750 800 850 900 950 1000 1050 Leaf Area (m2)

Figure 5. Fitted linear regression model for the 10 10 m stands of conifers, SAplot in relation to its LAplot. Figure 5. Fitted linear regression model for the ×10 × 10 m stands of conifers, SAplot in relation to its LAplot.

SA = 1.110 + 0.000312 LA 10 ) 2

Sapwood area Area (m Area area Sapwood 2

S 0.746821 0 Regression R-Sq 95.5% 95% CI R-Sq(adj) 93.2%

0 5000 10000 15000 20000 Leaf Area (m2)

Figure 6. Fitted linear regression model for the 60 × 60 m stands of conifers, SAplot in relation to its

LAplot.

Forests 2019, 10, 829 12 of 19

0.973, (<0.001) 1Correlation between 10 × 10 m stands, 2correlation between 60 × 60 m stands, 3correlation includes all stands

The sample set was then divided into the 10 × 10 m and the 60 × 60 m stands to fit two separate regression models for the conifer stands. The first model corresponded to the 10 × 10 m stands (Figure 5), and the latter model was fit for the 60 × 60 m stands (Figure 6). Regression analysis and ANOVA results suggest that the LAplot is a significant predictor of the SAplot (α = 0.05). However, the model’s R2pred for the 10 × 10 m stands denote inadequacy for future predictions, and the R2pred (38.37%) significantly differs from the R2adj. The regression model for the 60 × 60 m stands, however, show a better agreement between its coefficients of determination (Figure 6). Still the difference between R2adj and the R2pred was large (68.60%), but the model adequacy check gave enough evidence to support the decision for fitting a linear model to the 60 × 60 m data set. Both models have similar slopes that differ by just about 0.37 cm2.

0.45 SA = 0.03406 + 0.000349 LA ) 2 0.40

0.35

0.30

S 0.0317959 R-Sq 71.8% Sapwood area Area (m Area area Sapwood 0.25 R-Sq(adj) 64.7%

Regression 95% CI 0.20

700 750 800 850 900 950 1000 1050 Leaf Area (m2)

ForestsFigure2019, 10 5., 829Fitted linear regression model for the 10 × 10 m stands of conifers, SAplot in relation to its12 of 19 LAplot.

SA = 1.110 + 0.000312 LA 10 ) 2

Sapwood area Area (m Area area Sapwood 2

S 0.746821 0 Regression R-Sq 95.5% 95% CI R-Sq(adj) 93.2%

0 5000 10000 15000 20000 Leaf Area (m2)

Figure 6. Fitted linear regression model for the 60 60 m stands of conifers, SAplot in relation to its LAplot. Figure 6. Fitted linear regression model for the ×60 × 60 m stands of conifers, SAplot in relation to its 3.2.4.LA COVplot. Conﬁdence Intervals

The pairwise comparison of C.I. for COV was an extra analysis to support the applicability and reliability of the regression models, because the sample size for both data sets was not as large as desired when ﬁtting a linear regression model. Therefore, this analysis reinforced the suggested relationship between the scaling factors, no matter the sample size. Since most of the COVs were larger than 0.33, Payton’s equation (Equation (4)) was used to estimate the C.I. Table6 displays the obtained COVs and C.I. of SAplot, LAI, and LAplot.

Table 6. Coeﬃcient of variation (COV) results for the coniferous and deciduous stands’ SAplot, LAplot and LAI with their 95% conﬁdence intervals (95% C.I.s).

Site Type Variable COV 95% C.L.

SAplot 0.72 0.4963–1.6222 Coniferous LAplot 0.79 0.5943–1.9426 LAI 0.32 0.2438–0.7976 SAplot 1.46 0.6614–1.7885 Deciduous LAplot 1.82 0.7026–1.8999 LAI 0.17 0.1344–0.3633

For the coniferous data set, the LAplot, SAplot, and LAI confidence intervals were not significantly different; therefore, this suggested that there may be correlations amongst these scaling factors. For the deciduous data set, the LAplot and SAplot confidence intervals were not significantly different, while the LAI confidence interval was significantly different from the LAplot and SAplot confidence intervals. Therefore, this may be an indication of a correlation between LAplot and SAplot but there should be no correlation between SAplot and LAI in the deciduous data set. These results were in reasonable agreement with the results obtained with the Pearson’s correlation hypothesis test and the regression analysis. Forests 2019, 10, 829 13 of 19

3.3. Estimates of Error Propagation

The absolute error on SAplot estimates (∆SAplot) was calculated based on the rules of error propagation that are derived from a Taylor series [47]. Equation (8a) describes SAplot as the summation of each tree’s SA, thus, ∆SAplot will be given by the summation of each tree’s error contribution to SA:

T   X ∂SAplot ∂SAplot  SA =  D + sd0 ∆ plot  ∆ OBi ∆ i  (12)  ∂DOB 0  i=1 i ∂sdi

0 th where ∆DOBi and ∆sdi are the absolute errors on the i tree’s DOB and estimated sdi, respectively. The measurement of DOB were carefully veriﬁed by measuring the circumference of the trees at breast height on the same tree, 50 times. Two trees were measured in this exercise. The average error calculated was 0.0024 m. ± 0 For Picea glauca sd estimates (Equation (6)), ∆sdi will be given by:

∂sd0 ∆sd = i ∆D = 0.09∆D (13) i0 OBi OBi ∂DOBi

0 In the Populus tremuloides linear model (Equation (7)), ∆sdi is given by:

∂sd0 ∆sd = i ∆D = 0.24∆D (14) i0 OBi OBi ∂DOBi

0 Solving the partial derivatives in Equation (12) and substituting ∆sdi for Equation (13), the Picea glauca ∆SAplot is:

XT h i ∆SA = sd0∆D + ∆D 2sd0 0.09∆D (15) plot i OBi OBi − i OBi i=1 and for Populus tremuloides:

XT h i ∆SA = sd0∆D + ∆D 2sd0 0.24∆D (16) plot i OBi OBi − i OBi i=1

For the other three species studied here that have no linear relationship between average sapwood depth and diameter at breast height, ∆SAplot was estimated by the following equation:

∂SAplot ∂SAplot ∆SA = ∆SA + ∆T (17) plot sp ∂SAsp ∂T

Due to the direct method used in this study to measure T—the total number of trees inside a particular plot, it was considered an exact number (if some other approaches are used to indirectly estimate tree quantity, there may be an error associated with T); therefore,

∆SAplot = T∆SAsp (18) where ∆SAsp is the absolute error on the SA average value and is given by:

n   X ∂SAsp ∂SAsp ∂SAsp  SA =  D + sd + n ∆ sp  ∆ OBi ∆ ∆  (19)  ∂DOB ∂n  i=1 i ∂sd Forests 2019, 10, 829 14 of 19 where n is a constant; thus, the third term is null. The equation then becomes:   Xn  sd ∆D + D 2sd ∆sd   i OBi OBi − i  ∆SAsp =   (20)  n  i=1

According to previous research, the measure of sapwood depth with the microscopical analysis of wood anatomy gives an accuracy of 98% [48]. The sapwood depth measurements supporting this research were obtained from [37] and were measured using the microscopical method. It was estimated that the error on sd was related to the accuracy of the ocular scale of the microscope (with divisions of 1µ) and the ruler (with divisions of 1 mm) used to measure each core’s sapwood depth [37]. Thus, the instrument limit of error (ILE) was estimated as half of the smallest measuring increment of the 1 instrument (ruler). Hence, it was estimated that ∆sdi = ILE = 2 (1 mm) = 0.0005 m. ∆SAsp for the 2 ± Pinus contorta and Pinus banksiana sample set was 0.0002 m ; ∆SAsp for the Picea mariana sample set 2 was also 0.0002 m . Tables2 and3 report ∆SAplot for the coniferous and deciduous sites, respectively. Equation (21) is used to estimate the error on LAplot (∆LAplot):

∂LAplot ∂LAplot ∆LAplot = ∆LAI + ∆Aplot (21) ∂LAI ∂Aplot

∆LAI was 0.10 for the deciduous stands and 0.13 for the coniferous stands, while ∆A was ± ± plot given by:

∂Aplot ∆Aplot = ∆L (22) ∂L Thus, on the whole we obtain: ∆Aplot = 2L∆L (23) where L is the stand’s length, with ∆L = 2.79 m for deciduous stands and ∆L = 1.27 m for coniferous ± ± stands. The 10 10 m stands ∆L = 0.05 m. Solving Equation (23), a deciduous 60 60 m stand’s ∆A × ± × plot was 334.80 m2 and 138.26 m2 for the 415 m2 stand. A coniferous stand of 60 60 m had a ∆A of ± ± × plot 152.40 m2. Notice that ∆A of the stands named Conifer-11 and Conifer-12 (from Whitecourt) were ± plot 50.8 m2 and 44.45 m2, respectively (because they had a smaller surface area). For the 10 10 m ± ± × deciduous stands, ∆A equaled 1.00 m. The errors on LA estimates are given in Table4. Notice plot ± plot that the ∆LAplot became larger as the stand size increased, being more notorious in the larger stands. In addition, the contribution of ∆LAI to ∆Aplot was small, but still the size of the stand inﬂuenced the ﬁrst term of the Equation (21); however, if LAI increased in large stands, then ∆Aplot became large (e.g., Conifer-4, Deciduous-6). The prediction of SAplot for deciduous and coniferous stands using the obtained linear models, establishes that SAplot = f(LAplot). Thus, the error propagation on the linear models follows:

∂SAplot = ∆SAplot0 ∆LAplot (24) ∂LAplot

For the deciduous sites linear model, the ∆SAplot produced the following:

∆SA’plot = 0.00127∆LAplot (25)

For the coniferous linear model, the error on the SAplot was:

∆SA’plot = 0.000312∆LAplot (26) Forests 2019, 10, 829 15 of 19

4. Discussion and Conclusions Reliable estimates of tree and stand sapwood area are necessary in the calculation of other eco-hydrological parameters such as the actual transpiration from trees, for example. The error associated with the estimates of SA and SAplot will influence transpiration values if used to scale tree sap flow measurements. Since the accuracy of water balance components is crucial in this area of work, the authors believe that determining SA, SAplot, LA and LAplot propagated errors are fundamental to any such research and modelling. A number of studies have already highlighted the relevance of creating reliable allometric models, and have analyzed the error associated with their allometric correlations in order to support their reliability [26,49–51]. Regarding the allometric correlations, previous research [31] demonstrated that it was feasible to estimate a single tree’s sapwood area for Picea glauca and Populus tremuloides using estimates of sd from DOB. In this study, DOB was not an adequate sapwood area predictor of Pinus banksiana, Pinus contorta and Picea mariana species. As a result, scaling cross-sectional sapwood area to the stand-level was approached by combining the estimates of SA obtained with the regression models for Picea glauca and Populus tremuloides, and by using an average of the sapwood area of the remaining three species in a combined approach that scaled up sapwood area from a single tree to the stand scale. To the authors’ knowledge, the use of average sapwood area has not been previously studied as an option for scaling key, characteristic dimensions of trees. This combined approach gives reliable SAplot estimates that were significantly correlated to LAplot estimates derived from measurements of LAI. The final relationship (for both groups conifers and deciduous) allowed the development of linear models for predicting SAplot as function of LAplot. We attribute the reliability of the average sapwood area values to the careful and detailed methods that were used to measure a single tree’s sapwood depth, and that contributed to an error that was almost negligible [31]. This statement was supported by a more recent research study that reached similar conclusions regarding accurate measures of sapwood depth [52]. The errors associated with the SAplot estimates were mainly influenced by the size of the stand that in turn influenced the LAplot values—the larger the plot, the more the associated error with LAplot. This error (∆LAplot) had an impact on the estimates of SAplot having LAplot as predictor, and prevented stronger regression models by combining the 10 10 m and the 60 60 m coniferous stands. On the × × contrary, in the case of the Populus tremuloides, the addition of stands of a size in between 10 10 m × and 60 60 m would improve the model even further and may allow it to be applicable for use on × other data sets; and perhaps even allow the addition of the stand Deciduous-6 to the model. Another important point is that the reduction of error on leaf area estimates depends on accurate delimitation of the stands, and as the stand’s size increases, it is more complicated to retain accurate stand delimitation. Still, stands larger than 10 10 m are needed in order to limit discrepancies associated with the × footprint on LAI. Even though small stands give the smallest errors, there are large discrepancies between estimates of LAplot and SAplot due to the footprint influence. Consequently, we consider the analysis of the regression model for the 10 10 m stands helps to support the reliability of the × 60 60 m linear models (since both follow the same linear pattern and the slope is practically the × same). For future modelling, it might be suitable to add the slope as another predictor of sapwood area to the stand level, as suggested by [53] who determined that it is the slope that determines the correlation between stand-level leaf area and sapwood area. The authors recommend that at least for conifer trees, it is preferred to use stands that are bigger than 100 m2, and by trying to reduce the footprint influence on LAI derived from optical measurements. Furthermore, attention to the accuracy in the relationship between DOB and sd cannot be understated. In a simple computation to determine SA in the plots when simply using an average sapwood area (Equation (9)) for all species as opposed to Equation (9) only for Pinus contorta, Pinus banksiana and Picea mariana and the appropriate linear form of Equation (8) for Populus and Picea glauca, significant overestimates of SA were found (results not shown here). This in turn created an overestimate of the slope in the LA:SA models (as in Figures3–6) for each set of plots. Forests 2019, 10, 829 16 of 19

Recent studies have stressed the significant influence that scaled values of sapwood area can exert on final estimates of tree transpiration [54]. Thus, studies that have carefully analyzed the different methods to scale up tree morphological characteristics and their associated uncertainty, as in this paper, this should be a good reference for choosing reliable scaling methods in order to reduce the error that may be propagated into stand, ecosystem, or even watershed estimates. It is also recommended that the errors associated with estimates of tree transpiration for instance, should be determined and added into the overall error propagation equation. Some studies have, however, focused on taking into account the errors associated with only a section of their models [54,55]. For reliability purposes, the authors strongly recommend that one determine all the errors associated with every single parameter of a scaling model. In conclusion, these results establish the uniqueness of each species sapwood area, DOB and LA allometries. Thus, any attempt to use the reported correlations in this paper for other estimates should do so for similar site conditions, topographic and climatic characteristics. In this study, it was possible to aggregate cross-sectional sapwood area from the tree to the stand level in a mixed forest by differentiating between deciduous and coniferous groups of trees, and by combining two different approaches, with an error that is considered negligible. In terms of scaling, the results suggest that the SAplot and LAplot relationship is maintained at larger scales in this particular area and for the five studied species. There is still some variation that is not explained by the models. Thus, it is recommended that allometric characteristics of the trees, such as crown class, soil moisture or even vapor pressure deficit and maximum summer temperature as suggested by [56], could be introduced in the allometry by season, in order to observe what other factors influences SAplot.

Author Contributions: M.R.Q.-P. conducted the lab and field experiments, and together with C.V. conducted the data analysis and wrote the paper. Funding: This research was funded by the Alberta Ingenuity Fund, NSERC, Consejo Nacional de Ciencia y Tecnología, México, and Universidad Autónoma Metropolitana-Iztapalapa, México. Acknowledgments: Thanks are due to the Biogeoscience Institute of the University of Calgary, Alberta and field colleague David McAllister. Many thanks to the Reviewers for providing their insightful comments. Conflicts of Interest: The authors declare no conflict of interest.

References

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